Cauchy’s Integral Theorem – Local Version

By Sébastien Boisgérault, Mines ParisTech, under CC BY-NC-SA 4.0

September 30, 2019

Contents

Introduction

We derive in this document a first version of Cauchy’s integral theorem:

Theorem – Cauchy’s Integral Theorem (Local Version).

‌ Let \(f: \Omega \to \mathbb{C}\) be a holomorphic function. For any \(a \in \Omega,\) there is a radius \(r>0\) such that the open disk \(D(a, r)\) is included in \(\Omega\) and for any rectifiable closed path \(\gamma\) of \(D(a, r),\) \[ \int_{\gamma} f(z) \, dz = 0. \]

We will actually state and prove a slightly stronger version – one that does not require the restriction to small disks if \(\Omega\) is star-shaped.

In a subsequent document, we will prove an even more general result, the global version of Cauchy’s integral theorem. It will be applicable if \(\Omega\) is merely simply connected (that is “without holes”).

Integral Lemma for Polylines

Lemma – Integral Lemma for Triangles.

‌ Let \(f:\Omega \to \mathbb{C}\) be a holomorphic function. If \(\Delta\) is a triangle with vertices \(a,\) \(b\) and \(c\) which is included in \(\Omega\) \[ \Delta = \{ \lambda a + \mu b + \nu c \; | \; \lambda \geq 0, \, \mu \geq 0, \, \nu \geq 0 \, \mbox{ and } \, \lambda + \mu + \nu = 1 \} \subset \Omega \] and if \(\gamma = [a \to b \to c \to a]\) is an oriented boundary of \(\Delta\) then \[ \int_{\gamma} f(z) \, dz = 0. \]

Proof.

‌ Let \(a_0=a,\) \(b_0=b,\) \(c_0=c;\) consider the midpoints of the triangle edges: \[ d_0 = \frac{b_0 + c_0}{2}, \; e_0 = \frac{a_0 + c_0}{2}, \; f_0 = \frac{a_0 + b_0}{2}. \] The sum of the integrals of \(f\) along the four paths \([a_0 \to f_0 \to e_0 \to a_0],\) \([f_0 \to b_0 \to d_0 \to f_0],\) \([e_0 \to d_0 \to c_0 \to e_0],\) \([d_0 \to e_0 \to f_0 \to d_0]\) is equal to the integral of \(f\) along \(\gamma.\) By the triangular inequality, there is at least one path in this set, that we denote \(\gamma_1,\) such that \[ \left| \int_{\gamma_1} f(z) \, dz \right| \geq \frac{1}{4} \left| \int_{\gamma} f(z) \, dz \right|. \] We can iterate this process and come up with a sequence of paths \(\gamma_n\) such that \[ \left| \int_{\gamma_n} f(z) \, dz \right| \geq \frac{1}{4^n} \left| \int_{\gamma} f(z) \, dz \right|. \] Denote \(\Delta_n\) the triangles associated to the \(\gamma_n;\) they form a sequence of non-empty and nested compact sets. By Cantor’s intersection theorem, there is a point \(w\) such that \(w \in \Delta_n\) for every natural number \(n.\) The differentiability of \(f\) at \(w\) provides a complex-valued function \(\epsilon_w,\) defined in a neighbourhood of \(0,\) such that \(\lim_{h \to 0} \epsilon_w(h) = \epsilon_w(0) = 0\) and \[ f(z) = f(w) + f'(w) (z - w) + \epsilon_w(z - w) |z-w| \] Consequently, for any \(\epsilon > 0\) and for any number \(n\) large enough, \[ \left| \int_{\gamma_n} [f(z) - f(w) - f'(w) (z - w)] \, dz\right| \leq \epsilon \, \mathrm{diam} \, \Delta_n \times \ell(\gamma_n), \] where the diameter of a subset \(A\) of the complex plane is defined as \[ \mathrm{diam} \, A = \sup \, \{|z - w| \; | \; z \in A, \, w \in A\}. \] We have \(\ell(\gamma_n) = \ell(\gamma)/2^n\) and \(\mathrm{diam} \, \Delta_n = \mathrm{diam} \, \Delta_0 / 2^n.\) Additionally, \[ \int_{\gamma_n} f(w) \,dz = \int_{\gamma_n} f'(w) (z - w) \, dz = 0 \] since the functions \(z \in \mathbb{C} \mapsto f(w)\) and \(z \in \mathbb{C} \mapsto f'(w) (z-w)\) have primitives. Consequently, for any \(\epsilon > 0,\) for \(n\) large enough, \[ \frac{1}{4^n} \left| \int_{\gamma} f(z) \, dz \right| \leq \left| \int_{\gamma_n} f(z) \, dz\right| \leq \frac{1}{4^n}\epsilon \, \mathrm{diam} \, \Delta_0 \times \ell(\gamma), \] which is only possible if the integral of \(f\) along \(\gamma\) is zero. \(\blacksquare\)

Definition – Star-Shaped Set.

‌ A subset \(A\) of the complex plane is star-shaped if it contains at least one point \(c\) – a (star-)center, the set of which is called the kernel of \(A\) – such that for any \(z\) in \(A,\) the segment \([c, z]\) is included in \(A.\)

Lemma – Integral Lemma for Polylines.

‌ Let \(f: \Omega \to \mathbb{C}\) be a holomorphic function where \(\Omega\) is an open star-shaped subset of \(\mathbb{C}.\) For any closed path \(\gamma = [a_0 \to \dots \to a_{n-1} \to a_0]\) of \(\Omega,\) \[ \int_{\gamma} f(z) \, dz = 0. \]

Proof.

‌ Let \(c\) be a star-center of \(\Omega\) and define \(a_n = a_0;\) for any \(k \in \{0, \dots, n-1\},\) the triangle with vertices \(c,\) \(a_{k}\) and \(a_{k+1}\) is included in \(\Omega.\) Hence, by the integral lemma for triangles, the integral along the path \(\gamma_{k} = [c \to a_{k} \to a_{k+1} \to c]\) of \(f\) is zero. Now, as \[ \int_{\gamma} f(z) \, dz = \sum_{k=0}^{n-1} \int_{\gamma_{k}} f(z) \, dz, \] the integral of \(f\) along \(\gamma\) is zero as well. \(\blacksquare\)

Approximations of Rectifiable Paths by Polylines

To extend the integral lemma beyond closed polylines, we prove that polylines provide appropriate approximations of rectifiable paths:

Lemma – Polyline Approximations of Rectifiable Paths.

‌ Let \(\gamma\) be a rectifiable path. For any \(\epsilon_{\ell} > 0\) and \(\epsilon_{\infty} > 0,\) there is an oriented polyline \(\mu,\) with the same endpoints as \(\gamma,\) such that \[ \ell(\mu - \gamma) \leq \epsilon_{\ell} \; \mbox{ and } \; \forall \; t \in [0,1], \, |(\mu - \gamma)(t)| \leq \epsilon_{\infty}. \]
A 3-line approximation of the oriented unit circle.
A 3-line approximation of the oriented unit circle.
A 4-line approximation of the oriented unit circle.
A 4-line approximation of the oriented unit circle.
A 5-line approximation of the oriented unit circle.
A 5-line approximation of the oriented unit circle.

Proof – Polyline Approximations of Rectifiable Paths.

‌ Suppose that the path \(\gamma\) is continuously differentiable. Let \((t_0, \dots, t_n)\) be a partition of the interval \([0,1]\) and let \(\mu\) be the associated polyline: \[ \mu = [\gamma(t_0) \to \gamma(t_1)] \, |_{t_1} \, \cdots \, |_{t_{n-1}} \, [\gamma(t_{n-1}) \to \gamma(t_n)] \] The path \(\gamma\) and \(\mu\) have the same endpoints. The path \(\gamma\) may be considered as the concatenation \(\gamma = \gamma_1 \, |_{t_1} \, \dots \, |_{t_{n-1}} \, \gamma_n\) with the paths \(\gamma_{k}\) defined by \[ \forall \, k \in \{1, \dots, n\}, \, \forall t \in [0,1], \, \gamma_k(t) = \gamma\left(t_{k-1} + t (t_{k} - t_{k-1})\right), \] hence we have \[ \ell(\mu - \gamma) = \sum_{k=1}^{n} \int_0^1 \left| \gamma(t_{k}) - \gamma(t_{k-1}) - \gamma'_k(t)\right| \, dt. \] As \[ \gamma(t_{k}) - \gamma(t_{k-1}) = \int_{t_{k-1}}^{t_{k}} \gamma'(s) \, ds \] and \[ \begin{split} \gamma'_k(t) &= (t_{k} - t_{k-1})\gamma'\left(t_{k-1} + t (t_{k} - t_{k-1})\right) \\ &= \int_{t_{k-1}}^{t_{k}} \gamma'\left(t_{k-1} + t (t_{k} - t_{k-1})\right) \, ds, \end{split} \] we have the inequality \[ \ell(\mu - \gamma) \leq \int_0^1 \left[\sum_{k=1}^{n} \int_{t_{k-1}}^{t_{k}} |\gamma'(s) - \gamma'(t_{k-1} + t (t_{k} - t_{k-1}))| \,ds \right] dt \] The function \(\gamma'\) is by assumption continuous, and hence uniformly continuous, on \([0,1],\) therefore for any \(\epsilon > 0,\) there is a \(\delta(\epsilon) > 0\) such that, \(|\gamma'(s) - \gamma'(t)| < \epsilon\) whenever \(|s - t| < \delta(\epsilon).\) For any \(\epsilon_{\ell} > 0,\) for any partition \((t_0,\dots, t_n)\) such that \(|t_{k} - t_{k-1}| < \delta(\epsilon_{\ell})\) for any \(k \in \{1, \dots, n\},\) we have \[ \ell(\mu - \gamma) \leq \int_0^1 \left[\sum_{k=1}^{n} \int_{t_{k-1}}^{t_{k}} \epsilon_{\ell} \,ds \right] dt = \epsilon_{\ell}. \] For any \(\epsilon_{\infty} > 0,\) as \[ \forall \, t \in [0,1],\, |\mu(t) - \gamma(t)| \leq |\mu(0) - \gamma(0)| + \ell(\mu - \gamma) = \ell(\mu - \gamma), \] any partition \((t_0,\dots, t_n)\) such that \(|t_{k} - t_{k-1}| < \delta(\epsilon_{\infty})\) ensures that \[ \forall \; t \in [0,1], \, |(\mu - \gamma)(t)| \leq \epsilon_{\infty}. \] If \(\gamma\) is merely rectifiable, the same approximation process, applied to each of its continuously differentiable components provides the result. \(\blacksquare\)

Cauchy’s Integral Theorem

We finally get rid of the polyline assumption:

Theorem – Cauchy’s Integral Theorem (Star-Shaped Version).

‌ Let \(f: \Omega \to \mathbb{C}\) be a holomorphic function where \(\Omega\) is an open star-shaped subset of \(\mathbb{C}.\) For any rectifiable closed path \(\gamma\) of \(\Omega,\) \[ \int_{\gamma} f(z) \, dz = 0. \]

Proof.

‌ Let \(\epsilon > 0.\) Let \(r>0\) be smaller than the distance between \(\gamma([0,1])\) and \(\mathbb{C} \setminus \Omega.\) The set \[ K = \{z \in \mathbb{C} \, | \, d(z, \gamma([0,1])) \leq r \}, \] is compact and included in \(\Omega.\) Consequently, the restriction of \(f\) to \(K\) is bounded and uniformly continuous: there is a \(M>0\) such that \[ \forall \, z \in K, \; |f(z)| \leq M, \] and a there is a \(\eta_{\epsilon}>0\) – smaller than or equal to \(r\) – such that \[ \forall \, z \in K, \, \forall \, w \in \gamma([0,1]), \, |z - w| \leq \eta_{\epsilon} \, \Rightarrow \, |f(z) - f(w)| \leq \frac{\epsilon}{2 (\ell(\gamma)+1)}. \] Now, let \(\gamma_{\epsilon}\) be a closed polyline approximation of \(\gamma\) such that \[ \ell(\gamma_{\epsilon} - \gamma) \leq \frac{\epsilon}{2M} \; \mbox{ and } \; \forall \; t \in [0,1], \, |(\gamma_{\epsilon} - \gamma)(t)| \leq \eta_{\epsilon}. \] By construction, \(\gamma_{\epsilon}\) belongs to \(K,\) hence it is a closed path of \(\Omega.\) Therefore, the integral lemma for polylines provides \[ \int_{\gamma_{\epsilon}} f(z) \, dz = 0. \]

The rectifiable \(\gamma\) and \(\gamma_{\epsilon}\) have a decomposition into continuously differentiable paths associated to a common partition \((t_0,\dots,t_n)\) of the interval \([0,1]\): \[ \gamma = \gamma_1 \, |_{t_1} \, \cdots |_{t_n} \, \, \gamma_n \; \mbox{ and } \; \gamma_{\epsilon} = \gamma_{1\epsilon} \, |_{t_1} \, \cdots \, |_{t_n} \, \gamma_{n\epsilon} \]

The difference between the integral of \(f\) along \(\gamma\) and \(\gamma_{\epsilon}\) satisfies \[ \left| \int_{\gamma} f(z) \, dz - \int_{\gamma_{\epsilon}} f(z) \, dz \right| = \left| \sum_{k=1}^n \int_{0}^1 [ (f \circ \gamma_k) \gamma_k' - (f \circ \gamma_{\epsilon k}) \gamma_{\epsilon k}'](t) \, dt\right| \] Since for any \(k \in \{1,\dots, n\}\) \[ (f \circ \gamma_k) \gamma_k' - (f \circ \gamma_{\epsilon k}) \gamma_{\epsilon k}' = (f \circ \gamma_k - f \circ \gamma_{\epsilon k}) \gamma'_{k} + (f \circ \gamma_{\epsilon k}) (\gamma_k' - \gamma_{\epsilon k}'), \] we have \[ \begin{split} &\left| \int_{\gamma} f(z) \, dz - \int_{\gamma_{\epsilon}} f(z) \, dz \right| \\ &\leq \, \left| \sum_{k=1}^n \int_{0}^1 [ (f \circ \gamma_k - f \circ \gamma_{\epsilon k}) \gamma'_{k} ](t) \, dt\right| \\ &+ \, \left| \sum_{k=1}^n \int_{0}^1 [ (f \circ \gamma_{\epsilon k}) (\gamma_k' - \gamma_{\epsilon k}') ](t) \, dt\right| \end{split} \] and thus \[ \begin{split} \left| \int_{\gamma} f(z) \, dz\right| \leq &\, \max_{t \in [0,1]} |f(\gamma(t)) - f(\gamma_{\epsilon}(t))| \times \ell(\gamma) \\ & \, + \max_{t \in [0,1]} |f(\gamma_{\epsilon}(t))| \times \ell(\gamma_{\epsilon} - \gamma) \\ \leq &\, \frac{\epsilon}{2 (\ell(\gamma)+1)} \times \ell(\gamma) + M \times \frac{\epsilon}{2 M} \\ \leq &\, \epsilon. \end{split} \] As \(\epsilon > 0\) is arbitrary, the integral of \(f\) along \(\gamma\) is zero. \(\blacksquare\)

Consequences

Theorem – Cauchy’s Integral Formula for Disks.

‌ Let \(\Omega\) be an open subset of the complex plane and \(\gamma = c + r[\circlearrowleft]\) be an oriented circle such that the closed disk \(\overline{D}(c, r)\) is included in \(\Omega.\) For any holomorphic function \(f:\Omega \to \mathbb{C},\) \[ \forall \, z \in D(c,r), \; f(z) = \frac{1}{i2\pi} \int_{\gamma} \frac{f(w)}{w-z} dw. \]

Proof.

‌ Refer to the answers of exercise “Cauchy’s Integral Formula for Disks” \(\blacksquare\)

Corollary – Derivatives are Complex-Differentiable.

‌ The derivative of any holomorphic function is holomorphic.

Proof.

‌ Refer to the answers of exercise “Cauchy’s Integral Formula for Disks” \(\blacksquare\)

Theorem – Morera’s Theorem.

‌ Let \(\Omega\) be an open subset of \(\mathbb{C}.\) A function \(f:\Omega \to \mathbb{C}\) is holomorphic if and only if it is continuous and locally, its line integrals along rectifiable closed paths are zero: for any \(c \in \Omega,\) there is a \(r>0\) such that \(D(c,r) \subset \Omega\) and for any rectifiable closed path \(\gamma\) of \(D(c,r),\) \[ \int_{\gamma} f(z) \, dz = 0. \]

Proof.

‌ If \(f\) is holomorphic, then it is continuous and by Cauchy’s integral theorem, its line integrals along rectifiable closed paths are locally zero. Conversely, if \(f\) is continuous and all its line integrals along closed rectifiable paths are zero in some non-empty open disk \(D(c,r)\) of \(\Omega,\) \(f\) satisfies the condition for the existence of primitives in \(D(c,r).\) Any such primitive is holomorphic; since derivatives are complex-differentiable its derivative is holomorphic too and \(f\) is holomorphic in some neighbourhood of \(c.\) Since the initial assumption holds for any \(c \in \Omega,\) we can conclude that \(f\) is holomorphic on \(\Omega.\) \(\blacksquare\)

Theorem – Limit of Holomorphic Functions.

‌ Let \(\Omega\) be an open subset of \(\mathbb{C}.\) If a sequence of holomorphic functions \(f_n: \Omega \to \mathbb{C}\) converges locally uniformly to a function \(f:\Omega \to \mathbb{C},\) that is if for any \(c \in \Omega,\) there is a \(r>0\) such that \(D(c,r) \subset \Omega\) and \[ \lim_{n \to +\infty} \sup_{z \in D(c,r)} |f_n(z) - f(z)| = 0, \] then \(f\) is holomorphic.

Proof.

‌ The function \(f\) is continuous as a locally uniform limit of continuous functions. Now, let \(c \in \Omega\) and let \(r>0\) be such that \(D(c,r) \subset \Omega\) and the functions \(f_n\) converge uniformly to \(f\) in \(D(c,r).\) By Cauchy’s integral theorem, for any rectifiable closed path \(\gamma\) of \(D(c,r),\) the integral of \(f_n\) along \(\gamma\) is zero. Thus \[ \int_{\gamma} f(z) \, dz = \lim_{ n\to +\infty} \int_{\gamma} f_n(z) \, dz = 0. \] By Morera’s theorem, \(f\) is holomorphic. \(\blacksquare\)

Theorem – Liouville’s Theorem.

‌ Any holomorphic function defined on \(\mathbb{C}\) (any entire function) which is bounded is constant.

Proof.

‌ Let \(f:\mathbb{C} \to \mathbb{C}\) be a holomorphic function such that \(|f(z)| \leq \kappa\) for any \(z \in \mathbb{C}.\) Since derivatives are complex-differentiable, we may apply Cauchy’s integral formula for disks to the function \(f'\) and to the oriented circle \(\gamma = z + r[\circlearrowleft]\) for \(r>0\) and \(z \in \mathbb{C}.\) We have \[ f'(z) = \frac{1}{i2\pi} \int_{\gamma} \frac{f'(w)}{w-z} dw \] and by integration by parts, \[ f'(z) = \frac{1}{i2\pi} \int_{\gamma} \frac{f(w)}{(w-z)^2} dw, \] which yields by the M-L inequality \[ |f'(z)| \leq \frac{\kappa}{r}. \] This inequality holds for any \(r>0,\) thus \(f'(z) = 0\). Consequently, the zero function and \(f\) are both primitives of \(f'\); since the domain of \(f'\) is connected, these two primitives differ by a constant and thus \(f\) is constant. \(\blacksquare\)